To get a lower bound of \(d(\delta _1)\) , we need to use the following lemma in [1]}.
Lemma 5.1 ([1]}) 
For any \( \kappa \in (0,\frac{1}{4}) \) , if \( c^{\frac{\kappa }{2}}|Z|_{d} \le \frac{1}{4} \) , we have
\(0<\frac{[z_{11}^2]}{[z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2}\le \frac{1}{2}c^{-\kappa },\quad [z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2\ge 8c^{2\kappa }.\) 

Reviewing (REF ), we get \(c\le \epsilon _{0}^{\frac{3}{5}}e^{-\frac{r}{1-\chi }|2\pi k|^{{\nu }}}\) , \(|Z|_{R}\le D^{\prime } e^{3R(\pi |k|)^{{\nu }}}\) .
Then we set \(R=\frac{1}{8}\frac{\chi }{1-\chi }r\) , for \(\epsilon _0\)  sufficient small, we have
\( 0<\delta _1\le \epsilon _0^{\frac{1}{2}}e^{-r|k|^{{\nu }}}<4^{-1}D_{R}^{-1}|Z|_R^{-2} .\) 
Hence we can apply Lemma , and conjugate the system to the cocycle \( (\alpha , e^{b_0-\delta _1 b_1}+\delta _1^2P_1(\cdot )) \) .
Since \( c^{\frac{\chi }{2}}|Z|_R\le \frac{1}{4} \) , then we can apply Lemma REF , and get
\(\frac{[z_{11}^2]c}{[z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2}\le \frac{1}{2}\delta _1,\) 
hence we have
\(d(\delta _1)\ge c^{1-\chi }\cdot 8c^{2\chi }\cdot c^{1-\chi }\left( \frac{1}{2}-\frac{1}{8}c^{2-\chi }[z_{11}^2]\right) \ge \frac{9}{4}c^2.\) 
